David Dewhurst watched Monday as Gov. Rick Perry worked a phone bank for him in Dallas.

Ted Cruz spoke from Houston on Fox News on Monday in one of several late media appearances.

AUSTIN — A loud, nasty brawl to be Texas’ next U.S. senator comes to a probable conclusion Tuesday, and the outcome may shape the ferocity of Congress’ conservative wing come January.

GOP finalist David Dewhurst campaigned frantically Monday in three of the state’s four biggest media markets, dismissing an independent survey that showed opponent Ted Cruz with a 10-percentage-point lead as unreliable because it used an automated system to conduct interviews.

After tens of millions of dollars worth of ads, mostly negative, and months of campaigning, the race has taken on national import as a sign of how much the tea party, whose support Cruz happily claims, can sway the GOP. For the state Republican establishment, which has almost entirely aligned with Dewhurst, a Cruz victory could bring new challenges and other repercussions.

Cruz, 41, stands poised to become an immediate nationwide star if he becomes Texas’ first Hispanic senator.

The showdown follows more than a year of battling between Cruz and Dewhurst. They dispatched seven other Republicans in a late-May primary that had been delayed nearly three months by federal lawsuits over redistricting, with the extra time a key factor in Cruz’s being able to compete.

The winner will take on either Democrat Paul Sadler or Grady Yarbrough, who compete in a runoff of their own Tuesday, but the Republican nominee will almost certainly win in GOP-dominated Texas.

Cruz, the state’s former top appellate lawyer, relentlessly has framed the runoff as a choice between sending to Washington a “constitutional conservative” fighter or — as he’s cast Dewhurst, a third-term lieutenant governor — a conciliator with iron-poor blood and a spine deficit.

Dewhurst, after forfeiting a big early lead, has fought back, throwing at Cruz his huge checkbook and the staunch backing of every single top statewide officeholder — except Sen. Kay Bailey Hutchison, whose seat one of them will probably fill, Sen. John Cornyn and Attorney General Greg Abbott, Cruz’s old boss, who have all remained neutral.

Dewhurst says Cruz has shown poor judgment as an appellate lawyer and lacks the record of conservative achievement that the three-term lieutenant governor can tout. Campaigning in Dallas with Gov. Rick Perry and former primary rivals Tom Leppert and Craig James, Dewhurst promised to “shake up the status quo” in the nation’s capital.

He has cast Cruz as a creature of Washington-based conservative super PACs that have just one object — to shove aside the Senate’s current Republican leadership, including Texas Sen. John Cornyn, and install a clique now led by South Carolina Sen. Jim DeMint.

DeMint was among several national GOP stars, all non-Texans, who flocked to North Texas and the Houston area to stump for Cruz late last week.

Perry said DeMint’s faction wants to block Cornyn’s expected ascent to the post of the Senate’s No. 2 party leader, or whip, next year.

“There’s been millions of dollars pumped into the state of Texas by Club for Growth, by different groups that are more interested in who’s going to be the next majority whip in the U.S. Senate rather than what’s happening in Texas,” Perry said. “We all know that’s the truth.”

Cruz, sounding confident in his interview with Davis, seemed to peek over the horizon. If he wins the Republican nomination, he said, he’ll devote much effort this fall to trying to help elect “another five or six strong, constitutional conservatives” to the Senate.

“I intend, come Aug. 1, to try to help elect others so we can get the job done,” he said, citing plans to repeal President Barack Obama’s health care law and sharply curtail federal spending.

Dewhurst, who favors doing the same things, said the difference between him and Cruz isn’t agenda; it’s savvy.

“I know how to balance budgets, cut taxes. I want to go to Washington and turn Washington upside down. … It’s a mess,” he told reporters at a Chick-fil-A in south Austin. Dewhurst was there supporting the fast-food chain’s leaders, who are under fire for criticizing gay marriage.

Behind the scenes, both campaigns worked nonstop to drive turnout.

Conventional wisdom holds that a low turnout favors Cruz, while a bigger one swings the advantage gradually to Dewhurst, as casual voters more favorable to him play a bigger role.

In the state’s 15 counties with the most registered voters, about 288,000 people sent in mail ballots or voted early in person last week in the GOP runoff, according to figures from Texas Secretary of State Hope Andrade and Rice University political scientist Mark Jones.

That’s 84 percent of the GOP early vote from late May, so there may not have been the huge, dog-days-of-summer drop-off that some expected — although that won’t be clear until Tuesday’s results are in.

Jones said another dynamic is at work, potentially skewing conventional assessments of turnout’s impact.

“What that doesn’t account for is Cruz’s ability to peel off a significant share of Dewhurst voters and Dewhurst’s inability to bring really any — beyond a trivial number of people who supported Cruz in May — over to his side,” he said.

Parsing the latest polls and anecdotal reports, Jones said Cruz has “a strong advantage” among the most enthusiastic voters. Most probably voted by mail or in early voting, he said. Tuesday’s outcome hangs on who goes to the polls, he said.

“If Dewhurst is somehow able to motivate those casual November Republican voters to actually come out in the July heat and cast their vote, they’re on average going to favor him,” Jones said. “It’s doubtful that he’s going to be able to get a sufficient number [but] it’s possible.”

Staff writer Gromer Jeffers Jr. in Dallas contributed to this report.

Follow Robert T. Garrett on Twitter at @RobertTGarrett.

AT A GLANCE: Top races

Here’s a summary of the top races on the ballot in Tuesday’s runoffs:

FEDERAL

Senate: While Republicans decide the marquee GOP contest between David Dewhurst and Ted Cruz, Democrats will choose between Paul Sadler and Grady Yarbrough.

U.S. House: The Dallas area’s 33rd District showdown is among the top races, pitting Marc Veasey and Domingo Garcia in a Democratic battle for a new seat. In West Texas, Democrats will choose either Ciro Rodriguez or Pete Gallego to take on Republican freshman Rep. Francisco “Quico” Canseco in what will probably be the state’s hottest congressional race this fall. Elsewhere, Republicans will choose nominees in Central Texas and West Coast districts who will probably capture their seats in the fall.

STATE

Two Railroad Commission seats are on the Republican ballot, along with a state Supreme Court position. In the Dallas area, a key state Board of Education race pits former member Geraldine Miller against Gail Spurlock. Five state House districts feature runoffs, including two Dallas County GOP-leaning districts.

LOCAL

Dallas County has one contested judicial seat, as does Collin County. Rockwall Republican voters will choose a candidate for the county Commissioners Court.

TIPS FOR VOTING

Time: Polls are open from 7 a.m. to 7 p.m.

Eligibility: If you voted in a party primary May 29, you can vote in that party’s runoff but not the other party’s. If you skipped the primary, you can vote in either party’s runoff, but only one. How and whether you vote in a primary and runoff have no bearing on your vote in the fall general election.

Identification: You’ll need to show a voter registration card, a driver’s license or an official document with your address on it, such as a utility bill. You cannot be required to show a photo ID.

Registration: You must already be registered, and the deadline has passed.

Electioneering: It’s illegal to tout candidates or issues within a certain range of the polling place — and that includes wearing campaign gear to polling places. Election judges can stop you from voting if you are wearing campaign buttons or clothing. They have been known to make a person turn a shirt inside out before voting.

Printed material: You may bring notes or other materials with you. But you can’t distribute them or share them with other voters at the polling site.

Problems: Report any irregularities to the election judge on-site.

From staff reports

UPDATE: Business or faith? Or chicken?

Chick-fil-A has been at the center of the culture wars since its chief executive recently declared his opposition to gay marriage. But don’t tell that to Gov. Rick Perry.

Lt. Gov. David Dewhurst was asked Monday whether his visit to an Austin outlet of the fast-food chain was about business or faith, but the governor, joining him in Dallas to make voter-turnout calls, beat him to the punch.